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Discriminant 2:
Contribué par: Smith
(Auteur original: Tsygan)
The discriminant = b2 -4ac

The DISCRIMINANT is

the part of the

Quadratic Formula

under the √radical

The DISCRIMINANT determines how many solutions

the equation has.

 

The DISCRIMINANT also determines whether the solutions are:

 

 

rational (can be taken out of the radical) or

 

 

irrational (cannot be taken out of the radical)

The DISCRIMINANT is

the part of the

Quadratic Formula

under the √radical

√9 = 3
√7 =
√7
irrational
?
rational
?

The DISCRIMINANT determines how many solutions

the equation has.

 

The DISCRIMINANT also determines whether the solutions are

 

 

            (can be taken out of the radical) or

 

 

           (cannot be taken out of the radical)

The                             

is the part of the

Quadratic Formula

under the √radical

discriminant
?
√9 = 3
√7 =
√7

Negative radicals like  √-7 and √-4

will give you         real solution(s)

Positive radicals like √9 and √5

will give you        real solutions

Zero radicals like √0

will give you          real solution(s)

0
?
2
?
1
?

64

PERFECT SQUARES are √radicals

with WHOLE NUMBER (0,1,2,3,..) solutions

25

16

Simplify each perfect square below to a WHOLE number
=
=
 =
8

4

9

100

=
=
=

121

PERFECT SQUARES are √radicals

with WHOLE NUMBER (0,1,2,3,..) solutions

81

144

Simplify each perfect square below to a WHOLE number
=
=
 =

11

36

49

1

=
=
=
20 +√7

These are the 2 irrational solutions

 

The seven cannot be taken out

of the radical

Imperfect positive radicals

like √7 give 2 irrational solutions

20 ∓√7

Positive discriminants give 2 real solutions:

20 -√7

Perfect square positive radicals

like √9 give 2 rational solutions

20 +
20 +√9
20 ∓√9
20 -
20 -√9
20 +√7

These are the 2 irrational solutions

 

The seven cannot be taken out

of the radical

Imperfect positive radicals

like √7 give 2 irrational solutions

20 ∓√7

Positive discriminants give 2 real solutions:

20 -√7

Perfect square positive radicals

like √9 give 2 rational solutions

20 + 3
20 +√9
20 ∓√9
20 -√9
20 - 3
two                                  solutions

These are the 2 irrational solutions

 

The seven cannot be taken out

of the radical

20 +√7
irrational
?
20 ∓√7

Positive discriminants give 2 real solutions:

20 -√7

These are the 2 rational solutions

√9 reduced to 3

two                                 solutions
20 +√9
20 + 3
23
rational
?
20 ∓√9
20 -√9
20 - 3

17

Even though √7 is not a perfect square,

it will still give you two real solutions.

 

The solutions will be irrational

(you can't get rid of the radical √      )

CLICK OK

Even though √7 is not a perfect square,

it will still give you two real solutions.

 

The solutions will be irrational

(you can't get rid of the radical √      )

5

3

17

10

These IMperfect squares

will give you two real

IRRATIONAL solutions.

 

You cannot get rid of the

radicals

2 rational roots (no radicals)

If the discriminant is a perfect square,

then the quadratic will have

how many roots (solutions)?

 1 real root

2 irrational roots (with radicals)

no real roots (imaginary roots)

8 ∓√25

Perfect

Square

Example:

If the discriminant is +positive+, but NOT a perfect

square, then the quadratic will have

how many roots (solutions)?

2 rational roots (no radicals)

 1 real root

2 irrational roots (with radicals)

no real roots (imaginary roots)

Imperfect

Square

Example:

8 ∓√23

2 rational roots (no radicals)

 1 real root

2 irrational roots (with radicals)

no real roots (imaginary roots)

If the discriminant is ZERO,

then the quadratic will have

how many roots (solutions)?

Zero

Discriminant

Example:

8 ∓√0

If the discriminant is +positive+, but NOT a perfect

square, then the quadratic will have

how many roots (solutions)?

2 rational roots (no radicals)

 1 real root

2 irrational roots (with radicals)

no real roots (imaginary roots)

Imperfect

Square

Example:

8 ∓√13

2 rational roots (no radicals)

 1 real root

2 irrational roots (with radicals)

no real roots (imaginary roots)

If the discriminant is --NEGATIVE--,

then the quadratic will have

how many roots (solutions)?

NEGATIVE

Discriminant

Example:

8 ∓√-7

If the discriminant is a perfect square,

then the quadratic will have

how many roots (solutions)?

2 rational roots (no radicals)

 1 real root

2 irrational roots (with radicals)

no real roots (imaginary roots)

8 ∓√49

Perfect

Square

Example:

The

  End

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